Circles, Tangent Lines and Triangles

Objective

The figure below shows a semicircle, with diameter AB. Two tangent lines are drawn: one which touches the semicircle at B, the other at any point, C, on the semicircle. A line, CD, perpendicular to AB is dropped from the tangent point C. The two tangent lines intersect at the point T. The objectives of this project are to:

1. prove that the line AT bisects CD, and
2. illustrate the proof with a dynamic figure created with the Geometry Applet.

AB. A tangent line is drawn from B and from any point, C on the semicircle. The two tangent lines intersect at point T. Drop a line from C to point D on AB, such that CD is perpendicular to AB. Now, prove that a line drawn from A to T bisects CD." src="http://www.sciencebuddies.org/mentoring/project_ideas/CompSci_img010.gif">

Introduction

This is an ancient problem in geometry, posed and proved by the great Greek mathematician Archimedes in his Book of Lemmas.

The figure below shows a semicircle, with diameter AB. Two tangent lines are drawn: one which touches the semicircle at B, the other at any point, C, on the semicircle. A line, CD, perpendicular to AB is dropped from the tangent point C. The two tangent lines intersect at the point T. The goals of this project are to:

1. prove that the line AT bisects CD, and
2. illustrate the proof with a dynamic figure created with the Geometry Applet.

Note that the proposition holds true no matter where the point C falls on the semicircle. Click and drag on the point to see for yourself!

Figure 1: Prove that line AT bisects CD.

Notes on How to Manipulate the Diagram

The diagram is illustrated using the Geometry Applet (by kind permission of the author, see Bibliography). If you have any questions about the applet, send us an email at: scibuddy@sciencebuddies.org. With the help of the applet, you can manipulate the figure by dragging points.

In order to take advantage of this applet, be sure that you have enabled Java on your browser. If you disable Java, or if your browser is not Java-capable, then the figure will still appear, but as a plain, still image.

If you click on a point in the figure, you can usually move it in some way. The free points, usually colored red, can be freely dragged about, and as they move, the rest of the diagram (except the other free points) will adjust appropriately. Sliding points, usually colored orange, can be dragged about like the free points, except their motion is limited to either a straight line, a circle, a plane, or a sphere, depending on the point. Other points can be dragged to translate the entire diagram. If a pivot point appears, usually colored green, then the diagram will be rotated and scaled around that pivot point. (Note that figures will often use only one or two of the above types of points.)

You can't drag a point off the diagram, but frequently parts of the diagram will be moved off as you drag other points around. If you type r or the space key while the cursor is over the diagram, then the diagram will be reset to its original configuration.

You can also lift the figure off the page into a separate window. When you type u or return the figure is moved to its own window. Typing d or return while the cursor is over the original window will return the diagram to the page. Note that you can resize the floating window to make the diagram larger.

To learn how to use the Geometry Applet to create your own dynamic diagrams, see: Getting Started with the Geometry Applet

Terms, Concepts and Questions to Start Background Research

To do this project, you should do research that enables you to understand the following terms and concepts:

• tangent line,
• right triangles,
• semicircle of a right triangle,
• intersection of a line with parallel lines,
• similar triangles.

Questions:

• If a line is tangent to a circle, what do you know about the relation between the tangent line and the radius of the circle drawn to the tangent point?
• What is special about a triangle inscribed inside a semicircle, with one side of the triangle equal to the diameter of the semicircle?
• Given a right triangle, if you drop a line from the vertex of the right angle, perpendicular to the opposite base, you create two new right triangles within the larger one. What is special about these triangles?
• If a line intersects parallel lines, what do you know about the angles that are produced?

Bibliography

• The Math Forum at Drexel University has some good advice on how to build a mathematical proof:
  http://mathforum.org/library/drmath/view/54693.html
• There are many more examples in their FAQ section:
  http://mathforum.org/dr.math/faq/faq.proof.html
• For help with the programming the Geometry Applet, see:
  Geometry Applet How-to Pages
• The Geometry Applet was kindly provided to Science Buddies by its author, Professor David Joyce, who wrote it for illustrating an amazing online edition of Euclid's Elements. Here are some specific references that you may find useful for this project:
  • http://aleph0.clarku.edu/~djoyce/java/elements/bookIII/propIII18.html
  • http://aleph0.clarku.edu/~djoyce/java/elements/bookIV/propIV5.html
  • http://aleph0.clarku.edu/~djoyce/java/elements/bookVI/propVI8.html

Materials and Equipment

• For completing the proof manually, all you'll need is:
  • pencil,
  • paper,
  • compass, and
  • straightedge.
• For creating a dynamic diagram of the proof using the Geometry Applet, you will also need:
  • a computer,
  • a text editor (Notepad will work fine),
  • a Web browser program (e.g., Internet Explorer or Firefox).
• For a live demonstration along as part of your display, a laptop computer is recommended. Alternatively, if your school has a computer lab, talk to your teacher and see if you and other students doing computer science projects can arrange to display your science fair projects there.

Experimental Procedure

1. Do your background research. The Terms, Concepts and Questions section is a good place to start!
2. Organize your known facts (make a list!) Your list should include:
   1. the information given in the statement of the problem,
   2. relevant information from your background research, and
   3. relevant information from your knowledge of geometry.
3. Make sure you also write down a statement of the desired solution.
4. Try to build a list of the statements you need to prove in order to solve the problem. Remember that the goal of a proof is to construct a logical chain of steps leading from the given facts to the desired solution. Each step must be justified.
5. Constructing the proof does not have to be a one-way process, from beginning to end. You can also build backwards from the desired solution, and have your steps meet in the middle.
6. In addition to thinking logically, think visually!
   1. Remember that some of the facts you know about the problem will not be included in the original diagram which poses the problem. Finding ways to incorporate your known facts into the diagram may help you solve the problem.
   2. With many proofs, you also need to use your knowledge of geometry to build additional information into the diagram to solve the problem.
   3. Get yourself a few sheets of blank paper and try out your ideas as sketches.
7. Spend some time thinking about the problem and you should be able to come up with the proof.

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